What is a table? Finding the variables
A table puts facts in rows and columns so we can find them fast. In maths we use tables for two variables. A variable is something that can change.
First step: ask what changes. Example: you buy notebooks. The number of notebooks changes. The money you pay changes too. So the variables are number of notebooks and cost.
One variable is the one you choose (we call it x, the input). The other one follows from it (we call it y, the output). Always write the name and the unit in the table heading, like "cost (₹)".
Reading and comparing tables
Reading a table has three jobs.
- Look up: find the value of y for a given x. Go to the x column, then read across.
- Compare: put two rows side by side. Which is bigger? By how much?
- Find the pattern: as x goes up by 1, what does y do? Does it go up by the same amount each time?
In the 3D, every column of the table is a bar. A taller bar means a bigger y. Count the grid squares to read the same number you see in the table.
Building a table from a graph, a formula or a story
From a formula such as y = 2x: choose easy x values (1, 2, 3 …) and work out y for each. So x = 3 gives y = 6.
From a graph: pick x values on the horizontal axis. Go up to the line or the bar. Then go across to the vertical axis and read y. Write the pair in the table.
From a story: "A taxi charges ₹20 plus ₹10 for each km." Choose distances 1, 2, 3 km. Cost = 20 + 10 × km, so ₹30, ₹40, ₹50. Always check one row with the story.
Choose x values that are evenly spaced. This makes patterns easy to see.
Direct and inverse proportion in a table
Direct proportion: when x gets bigger, y gets bigger in the same ratio. Test: divide y by x in every column. If the answer is always the same number, the table is direct. That number is the constant of proportion. Then y = k × x.
Inverse proportion: when x gets bigger, y gets smaller in the same ratio. Test: multiply x by y in every column. If the answer is always the same, the table is inverse. Then x × y = k, so y = k ÷ x.
Neither: if y ÷ x changes and x × y changes, the table is not proportional. A table with y = x + 3 is like that.
A quick check: a direct table with x = 0 must have y = 0. Use this to reject tables like y = x + 3.
Try it: build your own table
Take 6 identical coins or buttons. Make 1 pile of 6, then 2 piles of 3, then 3 piles of 2, then 6 piles of 1. Write the number of piles (x) and coins in each pile (y). Multiply x × y. It is always 6. You just found an inverse table. Now use the 3D: pick "Inverse proportion" and check that your numbers have the same pattern. Before you click, predict: when x = 6, what will y be?
Key formulas and definitions
- Direct proportion: y ÷ x = k (same in every column), y = kx
- Inverse proportion: x × y = k (same in every column), y = k ÷ x
- Table from formula: choose x, compute y
- Not proportional: neither y ÷ x nor x × y is constant
Worked examples
1. A table shows x: 1, 2, 3 and y: 5, 10, 15. Is it direct proportion?
Step 1: divide y by x. 5 ÷ 1 = 5, 10 ÷ 2 = 5, 15 ÷ 3 = 5. Step 2: it is always 5. So it is direct proportion with k = 5, and y = 5x.
2. x: 1, 2, 4 and y: 12, 6, 3. What kind of table is it?
Step 1: try y ÷ x: 12, 3, 0.75. Not the same. Step 2: try x × y: 1 × 12 = 12, 2 × 6 = 12, 4 × 3 = 12. Always 12. So it is inverse proportion, y = 12 ÷ x.
3. Make a table for y = 3x + 1 with x = 0, 1, 2, 3.
x = 0: y = 3 × 0 + 1 = 1. x = 1: y = 4. x = 2: y = 7. x = 3: y = 10. Table: x 0, 1, 2, 3 and y 1, 4, 7, 10. y goes up by 3 each time.
4. A taxi costs ₹20 plus ₹10 per km. Build a table for 1, 2, 3, 4 km. Is the cost proportional to km?
Cost = 20 + 10 × km. For 1, 2, 3, 4 km: ₹30, ₹40, ₹50, ₹60. Check y ÷ x: 30, 20, 16.7, 15. Not constant. 0 km would still cost ₹20. So it is not direct proportion.
5. Eight workers finish a job in 6 days. Complete the table for 2, 4 and 12 workers if it is inverse.
x × y = 8 × 6 = 48. For 2 workers: y = 48 ÷ 2 = 24 days. For 4: 12 days. For 12: 4 days. Check: 2 × 24 = 4 × 12 = 12 × 4 = 48.
6. From a bar graph: the bar for 1 is 3 high, for 2 is 6, for 3 is 9, for 4 is 12. Write the table, the rule and the value at x = 10.
Table: x 1, 2, 3, 4 and y 3, 6, 9, 12. y ÷ x = 3 always, so y = 3x. At x = 10: y = 30.
Common mistakes
- Testing only one column. Test every column before you say "proportional".
- Saying y = x + 3 is direct because y grows when x grows. Direct needs the same ratio, not just growth.
- Mixing up the tests: divide for direct, multiply for inverse.
- Forgetting units in the heading, so 5 minutes and 5 hours get mixed up.